To find the fourth term of $(y - 3x)^6$, we can use the Binomial Theorem. The theorem states that:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Here, $a = y$, $b = -3x$, and $n = 6$. The general term (or the $(k+1)$-th term) of the expansion is given by:

$T_{k+1} = \binom{n}{k} a^{n-k} b^k$

Step 1: Identify the fourth term

For the fourth term, we need $k = 3$ (since the first term corresponds to $k = 0$). Thus, the fourth term is given by:

$T_4 = \binom{6}{3} y^{6-3} (-3x)^3$

Step 2: Calculate each part

1. Calculate the binomial coefficient:

   $\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$

2. Calculate $y^{6-3}$:

   $y^{6-3} = y^3$

3. Calculate $(-3x)^3$:

   $(-3x)^3 = -27x^3$

Step 3: Multiply everything together

$T_4 = 20 \cdot y^3 \cdot (-27x^3) = -540 y^3 x^3$

Final Answer

The fourth term of $(y - 3x)^6$ is:

$\boxed{-540 y^3 x^3}$

Would you like further details on any step, or have any other questions?

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1. What is the general term formula for a binomial expansion?
2. How to find the coefficient of a specific term in a binomial expression?
3. How does changing the sign inside the binomial affect the terms?
4. What would happen if we expand $(3x - y)^6$ instead?
5. How does the exponent impact the complexity of the expansion?

Tip: The power and coefficients in binomial expansions can be quickly identified by understanding patterns in Pascal's triangle.